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Mirrors > Home > MPE Home > Th. List > nfielex | Structured version Visualization version GIF version |
Description: If a class is not finite, then it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.) |
Ref | Expression |
---|---|
nfielex | ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 9202 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | eleq1 2817 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ Fin ↔ ∅ ∈ Fin)) | |
3 | 1, 2 | mpbiri 257 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin) |
4 | 3 | con3i 154 | . 2 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅) |
5 | neq0 4349 | . 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | sylib 217 | 1 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∅c0 4326 Fincfn 8970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-om 7877 df-en 8971 df-fin 8974 |
This theorem is referenced by: cusgrfi 29292 esumcst 33715 topdifinffinlem 36859 |
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